Chromatic Transversal Domatic Number of Graphs

The chromatic number χ(G) of a graph G is the minimum number of colours required to colour the vertices of G in such a way that no two adjacent vertices of G receive the same colour. A partition of V into χ(G) independent sets (called colour classes) is said to be a χpartition of G. A set S ⊆ V is called a dominating set of G if every vertex in V − S is adjacent to a vertex in S. A dominating set S of G is called a chromatic transversal dominating set (ctd-set) if S has non-empty intersection with every colour class of every χ-partition of G. The minimum order of a ctd-set of G is the chromatic transversal domination number of G and is denoted by γct(G). The chromatic transversal domatic number of a graph G is the maximum order of a partition of V into ctd-sets of G and is denoted by dct(G) . In this paper we obtain some bounds for dct(G) and characterize graphs attaining the bounds. Also we characterize uniquely colourable graphs with dct(G) = 1. Finally we obtain Nordhaus–Gaddum inequalities for dct(G) and characterize graphs for which dct(G) + dct(G) = p and p− 1. 640 L. Benedict Michael Raj, S. K. Ayyaswamy and I. Sahul Hamid Mathematics Subject Classification: 05C35